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Chapter 8: Problem 42

In Exercises \(41-48,\) solve the system using a graphing utility. Round all values to three decimal places. $$\left\\{\begin{array}{l} y=x^{3}-3 x+2 \\ y=4 x^{2}-1 \end{array}\right.$$

Short Answer

Expert verified
The solutions to the system of equations are the intersection points of the two graphs, each rounded to three decimal places. Verification is done by substituting the solutions into the original equations.

Step by step solution

01

Graph the first equation

Using a graphing utility, input the first equation \(y = x^{3} - 3x + 2\). This will produce a graph representative of the equation on the coordinate grid.
02

Graph the second equation

On the same graphing utility, input the second equation \(y = 4x^{2} - 1\). This graph will visually intersect with the first at points which represent the solutions to the system of equations.
03

Identify the points of intersection

Look at the graph to see where the two equations intersect. These intersection points represent the solution to the system of equations.
04

Round and verify

Round the x and y coordinates for each intersection point to three decimal places. Then, substitute these solutions into both original equations to assure they satisfy both equations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Utility
A graphing utility is a powerful tool used for visualizing mathematical equations. It's similar to a calculator but with a screen that can show graphs of equations in the coordinate plane. When solving systems of equations graphically, the utility helps us see where two or more equations intersect, which aids in finding common solutions. To use a graphing utility, follow these steps:

  • Input the equation into the utility. Each equation should be entered separately.
  • Adjust the viewing window to ensure the graph properly shows the behavior of the equations.
  • Observe the plotted graphs, focusing on essential features such as lines, curves, and intersection points.
These steps will help you interpret the graphs and extract meaningful information from them.
Intersection Points
The intersection points of graphs are crucial when solving a system of equations. An intersection point is where the graphs of the equations meet. For each intersection point, both equations have the same x and y coordinates. This implies that the point satisfies both equations simultaneously.

  • To find these points, look for places where the graphs cross each other on the coordinate grid.
  • The coordinates of these points are often where you'll find the numerical solution to the system.
  • Intersection points can be estimated visually and refined using a graphing utility's calculation feature to enhance accuracy.
Finding the intersection points is a foundational step in many applications, including real-world problem-solving scenarios.
Cubics and Quadratics
The given system includes a cubic and a quadratic equation. Cubic equations involve terms up to the third power (\(x^3\)), while quadratic equations involve terms up to the second power (\(x^2\)). Each type of polynomial has unique characteristics:

  • The cubic equation \(y = x^3 - 3x + 2\) can have up to three real roots, creating a curve which may turn twice.
  • The quadratic equation \(y = 4x^2 - 1\) forms a parabola, which typically has a "U" or an inverted "U" shape.

When working with these polynomials, their different shapes and types of roots mean intersection analysis may involve complex patterns, including multiple or no real intersection points. Recognizing these features is crucial for proper graphing and analysis.
Rounding Decimals
Rounding decimals is an essential skill in mathematics, especially when dealing with systems of equations that involve graphing. The goal is to simplify numbers to a certain degree of precision, making them easier to work with, without significantly losing accuracy.

When rounding to three decimal places:
  • Look at the fourth decimal place to decide if the third should round up or stay.
  • If the fourth decimal digit is 5 or greater, increase the third decimal by one.
  • If it's less than 5, leave the third decimal as is.
Rounding makes interpreting graphing results more straightforward and ensures consistency, especially when checking solutions by substituting them back into the original equations.

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Most popular questions from this chapter

A cab company charges \(\$ 4.50\) for the first mile of a passenger's fare and \(\$ 1.50\) for every mile thereafter. If it is snowing, the fare is increased to \(\$ 5.50\) for the first mile and \(\$ 1.75\) for every mile thereafter. All distances are rounded up to the nearest full mile. Use matrix addition and scalar multiplication to compute the fare for a \(6.8-\) mile trip on both a fair-weather day and a day on which it is snowing.

Find \(A^{2}\) (the product \(A A\) ) and \(A^{3}\) (the prod\(\left.u c t\left(A^{2}\right) A\right)\). $$A=\left[\begin{array}{rr}-4 & 0 \\\0 & 3\end{array}\right]$$

For the given matrices \(A, B,\) and \(C,\) evaluate the indicated expression. $$\begin{aligned}&A=\left[\begin{array}{rr}3 & 1 \\\2 & 5 \\\\-2 & 1\end{array}\right] ; \quad B=\left[\begin{array}{rr}-5 & -3 \\\1 & 6 \\\8 & 3\end{array}\right]\\\&C=\left[\begin{array}{rrr}2 & 1 & 1 \\\0 & -1 & 7 \\\3 & 0 & -3\end{array}\right] ; \quad C B+2 A\end{aligned}$$

In this set of exercises, you will use the method of solving linear systems using matrices to study real-world problems. Electrical Engineering An electrical circuit consists of three resistors connected in series. The formula for the total resistance \(R\) is given by \(R=R_{1}+R_{2}+R_{3},\) where \(R_{1}, R_{2},\) and \(R_{3}\) are the resistances of the individual resistors. In a circuit with two resistors \(A\) and \(B\) connected in series, the total resistance is 60 ohms. The total resistance when \(B\) and \(C\) are connected in series is 100 ohms. The sum of the resistances of \(B\) and \(C\) is 2.5 times the resistance of \(A\). Find the resistances of \(A, B\), and \(C\).

In this set of exercises, you will use the method of solving linear systems using matrices to study real-world problems. An electronics store carries two brands of video cameras. For a certain week, the number of Brand A video cameras sold was 10 less than twice the number of Brand B cameras sold. Brand A cameras cost \(\$ 200\) and Brand \(B\) cameras cost \(\$ 350 .\) If the total revenue generated that week from the sale of both types of cameras was \(\$ 16,750,\) how many of each type were sold?
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